what is the common ratio between 1/2 and -1/64

Question

what is the common ratio between  1/2 and -1/64

anonymous · Answer

i know its a geometric series but im confused on this..

campbell_st · Answer

to find the common ratio use 

$$\frac {T_{2}}{T_{1}}$$

so its 

$$\frac{-\frac{1}{64}}{\frac{1}{2}}$$

just evaluate it

anonymous · Answer

- 1/128

campbell_st · Answer

you just compare the terms.... realistically you should check the ratio is the same for the other terms 

$$\frac{T_{2}}{T_{1}} = \frac{T_{3}}{T_{2}} = \frac{T_{4}}{T_{3}}... $$

campbell_st · Answer

well the ratio is negative... but I don't think its -1/128

campbell_st · Answer

whats the rule for dividing by a fraction?

anonymous · Answer

the rule is  Turn the second fraction  upside-down 

Multiply the first fraction by that reciprocal
Simplify the fraction (if needed)

campbell_st · Answer

thats it

-1/64 x 2/1 =

anonymous · Answer

- 1/32

campbell_st · Answer

thats it

anonymous · Answer

my other qeustion states that to find 4 geometric means of this as well

so i just multiply to find it?

anonymous · Answer

for example 1/2 * -1/32

campbell_st · Answer

well isn't the geometric mean the nth root of the product of the 4 numbers...

anonymous · Answer

because the  qeustion is saying Write the geometric sequence that has 4 means between 1/2 and - 1/64

campbell_st · Answer

so find the geometric mean of 9, 3, 1, 1/3 

its $$\sqrt[4]{9 \times 3 \times 1 \times 1/3} = \sqrt[4]{9} = 1.73$$

anonymous · Answer

ok

campbell_st · Answer

I'm really unsure on the task... 

do they want 

1/2, -1/4, 1/8, -1/16, 1/32, -1/64 ?

anonymous · Answer

yes

campbell_st · Answer

and re reading the question it looks like the common ratio should be -1/2 

which allows you to generate the sequence with 4 terms between 1/2 and -1/64

so you know a = 1/2

and term 6 = -1/64

to find r is the task... here is the general formula for a term in a geometric sequence


$$T_{n} = ar^{n - 1} $$

so 

$$T_{6}:  -\frac{1}{64} = \frac{1}{2}\times r^{6 - 1}$$

solving for r is the task

divide both sides by 1/2 

$$-\frac{1}{32} = r^5..... or ...... r = \sqrt[5]{-\frac{1}{32}}....... r = - \frac{1}{2}$$

campbell_st · Answer

so the terms are 

$$\frac{1}{2}, \frac{1}{2} \times - \frac{1}{2}, - \frac{1}{4} \times - \frac{1}{2}.... $$